The development of a cognitively-diagnostic formative assessment of the early concept of angle
MetadataShow full item record
Students’ development of conceptual understandings is a central goal of mathematics education (CCSS-Mathematics, 2010). Such a challenging, yet ambiguous, goal cannot be achieved without empowering teachers with the knowledge and tools critical for their ability to adequately convey the content, and assess and interpret students’ performance. This study explicates the early concept of angle in terms of cognitive attributes that underlie children’s performance on angle-related mathematical tasks. The purpose of this study is to address the following gaps that were identified in the theory and research literature: the absence of a comprehensive description of the angle concept at early grades; the lack of developmental theory integrating the various aspects of angle competency and conceptualization near and at the grades in which the angle is first formally introduced; as well as the need for good formative assessments in mathematics. Specifically, the study investigates the early concept of angle – the angle-related knowledge, skills, and understandings appropriate and relevant for elementary students in grades 3 through 5 (the grade levels at which the concept is formally introduced). First, the synthesis of relevant cognitive models of conceptual development (e.g., Piaget, Greeno & Riley, Godino), theories of mathematical learning (e.g., Piaget, Mitchelmore & White, Sarama & Clements), Common Core Standards, and extant research was be used to describe the early concept of angle in terms of the cognitive attributes (knowledge, processes, and skills) that signify the mastery of the concept. Second, a Q-matrix theory and a combination of qualitative (item/task analysis and student interviews) and quantitative (Rule-Space Method) analyses guided the creation a formative assessment of the early concept of angle. The developed assessment was administered to 170 elementary students in grades 3, 4, and 5 from an elementary school in Southern California. A Rule-Space Methodology (Tatsuoka, 2009) was used to analyze the data and revise the Item and Attribute Sets. The use of the Rule-Space Methodology allowed for the identification, selection, and validation of the “best” items and “strongest” attributes to represent the concept form the conceptual, statistical, and psychometric standpoints. The classification procedure of the rule-space helped identify the various learning paths that students take towards the mastery of the concept. The study resulted in the definition of the early concept of angle in terms of 16 cognitive attributes and the development of a corresponding cognitively-diagnostic assessment of students’ competency levels with these attributes. In addition, the analysis of the collected empirical cross-sectional data revealed the various learning paths that students take towards the mastery of the concept. One “standard” and four “alternative” learning paths have been identified.